Use footway=crossing, together with highway=footway, to tag perpendicular sections of footway at a crossing point between two sidewalks tagged as separate way.

It is recommended to map the sidewalk as property on the street if the sidewalk is just attached with a kerb and not separated by a road verge or other barriers. If this alternative tagging is used there is no need for a perpendicular way and this tag. See Sidewalks for more details and a comparison of the two tagging approaches.